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Feedback cooling, measurement errors, and entropy production

The efficiency of a feedback mechanism depends on the precision of the measurement outcomes obtained from the controlled system. Accordingly, measurement errors affect the entropy production in the system. We explore this issue in the context of active feedback cooling by modeling a typical cold damping setup as a harmonic oscillator in contact with a heat reservoir and submitted to a velocity-dependent feedback force that reduces the random motion. We consider two models that distinguish whether the sensor continuously measures the position of the resonator or directly its velocity (in practice, an electric current). Adopting the standpoint of the controlled system, we identify the `entropy pumping' contribution that describes the entropy reduction due to the feedback control and that modifies the second law of thermodynamics. We also assign a relaxation dynamics to the feedback mechanism and compare the apparent entropy production in the system and the heat bath to the total entropy production in the super-system that includes the controller. In this context, entropy pumping reflects the existence of hidden degrees of freedom and the apparent entropy production satisfies fluctuation theorems associated to an effective Langevin dynamics.